Circles Class 10: Theorems & Tangent Properties

Q: What is a tangent to a circle?

A tangent to a circle is a straight line that touches the circle at exactly one point, called the point of contact or point of tangency. Unlike a secant, which crosses the circle at two points, a tangent never enters the interior of the circle.

Q: What is the relationship between a tangent and the radius at the point of contact?

The tangent at any point of a circle is always perpendicular to the radius drawn to that point of contact. This 90-degree relationship is the single most important fact in the chapter and is used to build a right triangle in almost every tangent-related problem.

Q: Are the two tangents drawn from an external point equal in length?

Yes. The lengths of the two tangents drawn from any external point to a circle are always equal. This follows from the congruence of the two right triangles formed by the radii, the tangents, and the shared line from the external point to the center.

Q: How many tangents can be drawn to a circle from a given point?

It depends on the point's position. From a point inside the circle, no tangent can be drawn. From a point on the circle, exactly one tangent can be drawn. From a point outside the circle, exactly two tangents can be drawn, and they are equal in length.

Q: What are the most common mistakes in Class 10 Circles?

Common mistakes include forgetting that the tangent is perpendicular to the radius (not the diameter or a chord in general), applying the equal-opposite-sides rule to quadrilaterals that don't actually circumscribe a circle, confusing a tangent with a secant, and misremembering how many tangents can be drawn depending on the point's position relative to the circle.

Circles Class 10 Maths – important theorems and tangent properties

Circles Class 10: Theorems & Tangent Properties Explained (2026 Guide)

📅 Updated Sep 14, 2026 ⏱ 12 min read 📘 CBSE / NCERT Class 10, Chapter 10 ✍️ MyTestSeries Expert Team

Circles Class 10 — tangent and radius theorems explained by MyTestSeries

Circles is one of the shortest chapters in Class 10 Maths — just two formal theorems — but it's deceptively rich in applications. Nearly every problem in this chapter is really the same right triangle in disguise: a radius, a tangent, and the line joining an external point to the center, related through one simple fact — the tangent is always perpendicular to the radius at the point where it touches.

This guide explains both theorems clearly, shows how they combine to prove that tangents from an external point are always equal in length, and works through six solved examples — including the classic circumscribed-quadrilateral problem — followed by the specific mistakes that quietly cost marks.

2Core Theorems

6Solved Examples

7Common Mistakes

1Quick Formula Table

📋 Table of Contents

What Is a Tangent? Tangent vs Secant
Theorem 1 — Tangent Perpendicular to Radius
Theorem 2 — Equal Tangent Lengths
Circumscribed Quadrilaterals
6 Fully Solved Examples
Common Mistakes Students Make
Quick Formula Recap Table
How to Revise This Chapter Effectively
Frequently Asked Questions
Related Test Series & Practice Sets

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1. What Is a Tangent? Tangent vs Secant

A tangent to a circle is a straight line that touches the circle at exactly one point — called the point of contact — without ever crossing into the circle's interior. This is different from a secant, which intersects the circle at two distinct points.

Point's Position	Number of Tangents Possible
Inside the circle	0 (no tangent can be drawn)
On the circle	1 (exactly one tangent, at that point)
Outside the circle	2 (always equal in length — see Theorem 2)

2. Theorem 1 — Tangent Perpendicular to Radius

The tangent at any point of a circle is perpendicular to the radius through the point of contact. If O is the center and P is the point of contact, then OP ⊥ the tangent line at P. This single fact builds a right angle you can use in almost every problem in the chapter.

💡 Quick check: This perpendicularity is exactly what lets you apply the Pythagoras theorem whenever a tangent, a radius, and the line to an external point are all part of the same diagram.

3. Theorem 2 — Equal Tangent Lengths

The lengths of the two tangents drawn from an external point to a circle are equal. If PA and PB are tangents from external point P, touching the circle at A and B, then PA = PB.

This follows directly from Theorem 1: triangles OAP and OBP both have a right angle (at A and B respectively), share the hypotenuse OP, and have equal radii OA = OB — so by the RHS congruence rule, the triangles are congruent, which makes PA = PB.

Just two theorems, but dozens of ways they get combined in exam questions. Practise recognising the pattern with the 7-Day Demo Test Series — instant, explained solutions after every attempt.

4. Circumscribed Quadrilaterals

If a quadrilateral ABCD circumscribes a circle (all four sides touch the circle), then: AB + CD = BC + AD Direct consequence of Theorem 2 — each side splits into two tangent segments from its corners, and equal tangent pairs cancel out neatly when opposite sides are added.

5. Six Fully Solved Examples

Example 1 — Finding the angle at the center

Two tangents PA and PB are drawn from an external point P to a circle with center O, such that ∠APB = 80°. Find ∠AOB.

In quadrilateral OAPB: ∠OAP = 90°, ∠OBP = 90° (tangent ⊥ radius), and ∠APB = 80°.
Sum of angles in a quadrilateral = 360°: ∠AOB = 360° − 90° − 90° − 80° = 100°

Example 2 — Length of a tangent

Find the length of the tangent drawn from an external point 25 cm from the center of a circle of radius 7 cm.

Using the right triangle formed by the radius, tangent, and the line to the external point:
PA² = PO² − OA² = 25² − 7² = 625 − 49 = 576 → PA = √576 = 24 cm

Example 3 — Two concentric circles

A chord of a larger circle (radius 5 cm) touches a smaller, concentric circle (radius 3 cm). Find the length of the chord.

The chord is tangent to the smaller circle, so the perpendicular distance from the common center to the chord equals the smaller radius, 3 cm.
Half the chord = √(5² − 3²) = √(25−9) = √16 = 4 cm. Full chord length = 8 cm.

Example 4 — Circumscribed quadrilateral

Quadrilateral ABCD circumscribes a circle, with AB = 6 cm, BC = 7 cm, and CD = 4 cm. Find AD.

Using AB + CD = BC + AD: 6 + 4 = 7 + AD → 10 = 7 + AD → AD = 3 cm

Example 5 — Proving equal tangent lengths

P is an external point, and PA, PB are tangents to a circle with center O, touching at A and B. Prove that PA = PB.

In triangles OAP and OBP: ∠OAP = ∠OBP = 90° (tangent ⊥ radius), OP is common (shared hypotenuse), and OA = OB (both radii).
By RHS congruence, △OAP ≅ △OBP. Therefore PA = PB (corresponding parts of congruent triangles).

Example 6 — Finding the angle between a tangent and the line to the center

Two tangents PA and PB are drawn from point P to a circle with center O, such that ∠APB = 60°. Find ∠AOP.

Since △OAP ≅ △OBP (Example 5), OP bisects ∠APB, so ∠APO = 30°.
In right triangle OAP: ∠OAP = 90°, ∠APO = 30°, so ∠AOP = 180° − 90° − 30° = 60°

6. Common Mistakes Students Make

⚠️ Mistake 1 — Forgetting the tangent is perpendicular to the radius specifically. Not to the diameter in general, and not to a chord unless that chord happens to pass through the point of contact along the radius direction.

⚠️ Mistake 2 — Using PA² = PO² − OA² without confirming the right-angle setup. This formula only works because ∠OAP = 90° — always state that reasoning before applying it.

⚠️ Mistake 3 — Applying the circumscribed-quadrilateral rule to any quadrilateral. AB+CD=BC+AD only holds when the quadrilateral genuinely circumscribes the circle — meaning all four sides are tangent to it, not just some.

⚠️ Mistake 4 — Confusing tangent with secant. A tangent touches at exactly one point; a secant crosses at two. Mixing these up leads to setting up the wrong triangle entirely.

⚠️ Mistake 5 — Misremembering the tangent-count rule. Zero tangents from inside, one from on the circle, two from outside — this small fact is tested directly in MCQs more often than expected.

⚠️ Mistake 6 — Forgetting the quadrilateral angle sum in OAPB-style problems. The four angles of quadrilateral OAPB always sum to 360° — skipping this step is the main reason Example 1's method goes wrong.

⚠️ Mistake 7 — Not stating RHS congruence explicitly when proving PA = PB. Many students know the result but skip naming the congruence criterion, which costs presentation marks in proof-based questions.

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7. Quick Formula Recap Table

Concept	Formula / Rule
Tangent ⊥ radius	OP ⊥ tangent line at point of contact P
Equal tangent lengths	PA = PB (tangents from external point P)
Tangent length formula	PA² = PO² − r² (r = radius)
Circumscribed quadrilateral	AB + CD = BC + AD
Tangent count	0 (inside), 1 (on circle), 2 (outside)
OP bisects ∠APB	By congruence of △OAP and △OBP

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8. How to Revise This Chapter Effectively

📖 Draw the OAPB quadrilateral by default for any two-tangent problem. Labelling the center, both points of contact, and the external point immediately reveals two right angles and often makes the rest of the problem obvious.

✍️ Learn the RHS congruence justification by heart. Right angle, hypotenuse, one pair of equal sides — this exact phrase is expected whenever you're asked to prove PA = PB.

🧮 Check that a quadrilateral genuinely circumscribes a circle before using the opposite-sides rule. The question should say all four sides touch the circle — don't assume it.

🎯 Treat every tangent-length question as a Pythagoras question first. Identify the right angle, name the hypotenuse, and the rest is direct substitution.

🔁 Revisit this page before your unit test. Bookmark it, and use the Quick Formula Recap Table (Section 7) as your final five-minute revision before the exam.

📚 Also Useful on MyTestSeries

🌐 Trusted External References

NCERT Official Website — download the official Class 10 Maths textbook
CBSE Academic — official syllabus & sample papers
Tangent Lines to Circles — Wikipedia — broader mathematical background
Circle — Wikipedia — general properties and history

9. Frequently Asked Questions

What is a tangent to a circle?

A straight line that touches a circle at exactly one point, without crossing into the circle's interior — different from a secant, which intersects at two points.

What is the relationship between a tangent and the radius at the point of contact?

The tangent is always perpendicular to the radius drawn to the point of contact — a 90° relationship used throughout the chapter.

Are the two tangents drawn from an external point equal in length?

Yes, always. This follows from the congruence (by RHS) of the two right triangles formed by the radii, the tangents, and the shared line to the center.

How many tangents can be drawn to a circle from a given point?

Zero from a point inside the circle, one from a point on the circle, and two (always equal in length) from a point outside the circle.

What are the most common mistakes in Class 10 Circles?

Forgetting the tangent is perpendicular specifically to the radius, misapplying the circumscribed-quadrilateral rule, confusing tangents with secants, and misremembering how many tangents are possible from each type of point.

Conclusion: Two Theorems, One Right Triangle

Every result in this chapter — the angle-sum trick in OAPB, the tangent-length formula, the circumscribed-quadrilateral rule — comes back to the same right angle between a tangent and its radius. Once that single fact is second nature, spotting the right triangle in any Circles question becomes almost automatic.

Rework the six solved examples above without checking the answers first, always name the congruence criterion when proving equal tangent lengths, and default to drawing the OAPB quadrilateral whenever two tangents from the same external point appear. That habit is what turns two short theorems into consistent exam marks.

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Tags: Circles Class 10, Tangent to a Circle, Tangent Perpendicular to Radius, Equal Tangent Lengths, Circumscribed Quadrilateral, CBSE 2026, NCERT Class 10 Maths Chapter 10

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